Explaining Individual Response using AggregatedData

Bram van Dijk∗

Econometric InstituteTinbergen Institute

Erasmus University Rotterdam

Richard PaapEconometric Institute

Erasmus University Rotterdam

ECONOMETRIC INSTITUTE REPORT EI 2006-05

Abstract

Empirical analysis of individual response behavior is sometimes limited due tothe lack of explanatory variables at the individual level. In this paper we put forwarda new approach to estimate the effects of covariates on individual response, wherethe covariates are unknown at the individual level but observed at some aggregatedlevel. This situation may, for example, occur if the response variable is available atthe household level but covariates only at the zip-code level.

We describe the missing individual covariates by a latent variable model whichmatches the sample information at the aggregate level. Parameter estimates canbe obtained using maximum likelihood or a Bayesian approach. We illustrate theapproach estimating the effects of household characteristics on donating behaviorto a Dutch charity. Donating behavior is observed at the household level, while thecovariates are only observed at the zip-code level.

JEL Classification: C11, C51Keywords: aggregated explanatory variables, mixture regression, Bayesian analy-sis, Markov Chain Monte Carlo

∗We thank Dennis Fok, Philip Hans Franses, Rutger van Oest, Bjorn Vroomen and participantsof seminars at the Institute of Advanced Studies in Vienna, the Universite Catholique de Louvain inLouvain-la-Neuve, NAKE research day in Amsterdam, and Facultes Universitaires Notre-Dame de laPaix in Namur, for helpful comments. Corresponding author: Tinbergen Institute, Erasmus UniversityRotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands, e-mail: avandijk@few.eur.nl, phone:+31-10-4088943, fax: +31-10-4089162.

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1 Introduction

Empirical analysis of individual behavior is sometimes limited due to the lack of explana-

tory variables at the individual level. There may be various reasons why individual-level

explanatory variables are not available. When using individual revealed preference data,

information about explanatory variables may simply not be available as databases cannot

be properly linked. When using surveys, one may be confronted with a missing question

concerning an important explanatory variable. It may also be the case that respondents

interpreted the relevant question the wrong way which makes the explanatory variable

unusable.

In some cases it may be possible to obtain information on explanatory variables at

some aggregated level. For example, if the zip code of households is known, one may

obtain aggregated information on household characteristics, like income and family size,

at the zip-code level. This zip-code level information is usually obtained through surveys.

The aggregated information of the variables is summarized in marginal probabilities which

reflect the probability that the explanatory variable lies in some interval (income, age) or

category (gender, religion) for a household in that zip-code region.

The goal of the current paper is to estimate the effects of covariates on individual

response when the covariates are unobserved at the individual level but observed at some

aggregated level. This problem is related to the literature on ecological inference, see, for

example, Wakefield (2004) for an overview. The main difference with regular ecological

inference problems is that we observe individual responses, whereas in ecological inference

one also has to rely on aggregated information on the response variable. The extra

information on individual responses may help us to overcome certain identification issues

in ecological inference.

There are several studies in economics which link individual and aggregated data,

see, for example, Imbens and Lancaster (1994) and van den Berg and van der Klaauw

(2001). The difference of these studies with our problem is that they assume that both

individual-level data and aggregated data is available. The aggregated data is assumed to

be more reliable and is used to put restrictions on the individual-level data. Our problem

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bears more similarities with symbolic data analysis, see Billard and Diday (2003) for an

overview. Symbolic data analysis also deals with aggregated explanatory variables and

dependent variables at an individual level. The motivation for the use of aggregated

data is however different. Aggregation is pursued to summarize large datasets. Therefore

the form of the aggregated information is different and represents, for example, intervals

instead of marginal probabilities.

As far as we know, the only paper that comes close to our situation is Steenburgh

et al. (2003). The motivation in this paper is however different from ours. They use

zip-code information to describe unobserved heterogeneity in the individual behavior of

households instead of estimating the effects of covariates on behavior.

To deal with our specific problem we propose in this paper a new approach to estimate

the effects of covariates on individual response, where the covariates are unknown at the

individual level but observed at some aggregated level. We add to the model describing

the individual responses a latent variable model for the explanatory variables. This latent

variable model describes the missing explanatory variables in such a way that it matches

the sample information at the aggregated level. In case of one explanatory variable, the

model simplifies to a mixture regression. A simple simulation experiment shows that

this new approach outperforms in efficiency the standard method, where one replaces the

missing explanatory variables by the observed marginal probabilities at the aggregated

level.

Parameter estimates of the response model can be obtained using Simulated Maximum

Likelihood [SML] or a Bayesian approach. Given the computational burden of SML, the

latter approach may be more convenient. To obtain posterior results, one can use a Gibbs

sampler with data augmentation. The unobserved explanatory variables are sampled

alongside the model parameters. Conditional on the sampled explanatory variables, one

can use a standard Markov Chain Monte Carlo [MCMC] sampler for the model describing

individual response.

The outline of the paper is as follows. In Section 2 we provide a simple introduction

into the problem and perform a small simulation experiment to illustrates the merits of our

approach. In Section 3 we generalize the discussion to a more general setting. Parameter

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estimation is discussed in Section 4. In Section 5 we illustrate our approach estimating

the effects of household characteristics on donating behavior to a Dutch charity. We use

aggregated information on household characteristics at the zip-code level to explain the

individual response of households to a direct mailing by the charity. Finally, Section 6

concludes.

2 Preliminaries

To illustrate the benefits of our new approach, we start the discussion with a simple

example. We consider a linear regression model with only one explanatory variable. The

explanatory variable xi can only take the value 0 or 1, for example, a gender dummy. Let

the observed response of individual i, yi, be described by

yi = α + βxi + εi, (1)

where α is an intercept parameter and where β describes the effect of the 0/1 dummy

variables xi on yi for i = 1, . . . , N . The error term εi is assumed to be normally distributed

with mean 0 and variance σ2. We assume that xi is unobserved at the individual level

but that we have aggregated information on xi, for example, at the zip-code level. This

aggregated information is summarized by pi = Pr[xi = 1] for i = 1, . . . , N .

A simple approach to estimate β is to regress yi on pi instead of xi. The error term of

this regression equals

ηi = (xi − pi)β + εi. (2)

The OLS estimator is consistent if E[piηi|pi] = 0. As

E[piηi|pi] = E [pi × ((xi − pi)β + εi)|pi] = E [pi(xi − pi)β|pi] + E[piεi|pi]

= E [pixiβ|pi]− E[p2

i β|pi

]+ E[piεi|pi] (3)

this condition is fulfilled if E[piεi|pi] = 0 and E[xi|pi] = pi. Although this OLS estimator

is consistent, it is clear from (2) that the error term is heteroskedastic, and hence the OLS

estimator is not efficient. Hence, one may opt for a GLS estimator.

An alternative approach to use the aggregated information to estimate β is to con-

sider a mixture regression, see Quandt and Ramsey (1978); Everitt and Hand (1981);

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Titterington et al. (1985). To describe the response variable yi we consider a mixture of

two regression models where in the first component the xi variable is 1 and in the second

component xi equals 0. The mixing proportion is pi which is known but may be different

across individuals. Hence, the distribution of yi is given by

yi ∼

N(α + β, σ2) with probability pi

N(α, σ2) with probability (1− pi).(4)

The parameters α and β can be estimated using maximum likelihood [ML]. ML estimates

can easily be obtained using the EM algorithm of Dempster et al. (1977).

To illustrate the efficiency gain of the mixture approach we perform a simulation

study. For N = 1, 000 individuals we simulate 0/1 xi values according to Pr[xi = 1] =

pi. We use different simulation schemes for pi. We either allow the value of pi to be

different across individuals, or we impose that groups of individuals have the same value

for pi corresponding to the idea that these individuals live the same zip-code region.

Furthermore, we allow the range of possible values for pi to be different. We sample pi from

U(0.2, 0.4) or U(0.01, 0.99). The values of yi are generated according to yi = 1 + 2xi + εi

with εi ∼ N(0, σ2) and σ2 = 1.

We estimate the β parameter using four approaches. In the first approach we estimate

β using a linear regression model where we include the true xi as explanatory variables.

In practice this solution is of course not feasible but it allows us to compute the efficiency

loss due to using explanatory variables at an aggregated level. In the second approach we

consider an OLS estimator in a linear regression model with pi as explanatory variable.

The third approach uses a GLS estimator in the same linear regression model. The GLS

weights are based on (3) and are computed using the true value of β and σ2. In practice

these parameters are of course unknown but the simulation results already show that

accounting for heteroskedasticity using the true values does not compensate the efficient

loss of the OLS estimator. In the last approach we consider the mixture solution as in

(4).

Table 1 displays the efficiency loss in the estimator for β for the last three estimation

approaches compared to using full information. Simulation results are based on 1000

replications. The efficiency loss is computed using the root mean squared error of the

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Table 1: Efficiency loss of using aggregated data withrespect to using full information for the three estimators

Distribution Number of Efficiency Lossof pi pi

a OLS GLS MixtureU(0.20, 0.40) 1,000 90.5% 90.5% 33.3%U(0.01, 0.99) 1,000 50.4% 49.8% 24.1%U(0.20, 0.40) 100 90.4% 90.4% 32.4%U(0.01, 0.99) 100 52.5% 52.2% 23.0%U(0.20, 0.40) 10 92.4% 92.3% 31.5%U(0.01, 0.99) 10 62.4% 62.1% 23.0%U(0.20, 0.40) 2 96.6% 96.6% 33.5%U(0.01, 0.99) 2 73.9% 73.9% 31.3%a Number of different pi values drawn from the uni-

form distribution. Number of individuals is 1,000.

estimates as all estimators are consistent. Several conclusions can be drawn from the table.

First of all, the mixing approach outperforms the other two estimators. Secondly, the GLS

estimator hardly improves upon the OLS estimator, indicating that heteroskedasticity is

not the main cause of the efficiency loss of the OLS estimator. Thirdly, all estimators

perform better when the range in possible values of pi is larger, which is not a surprise as

a large variation in pi provides more information about the slope parameter. Finally, the

estimators perform better when there are less individuals with the same pi value. The

mixing approach however seems hardly affected by the number of individuals with the

same value for pi.

To illustrate the effects of the efficiency loss, we display in Figure 1 the density of β

for the full information estimator, the OLS estimator and the mixture approach, where

we use the simulation settings as in the first line of Table 1. The graph clearly illustrates

the superiority of the mixture approach.

As already indicated by our simulation results, a GLS estimator does not compensate

the efficiency loss due to aggregation of the explanatory variables. A second reason why

the GLS estimator is not useful, is that constructing a feasible GLS estimator is often not

possible if we have more than one explanatory variable. Consider, for example, the case

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−1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1

2

3

4

5

Density

Mixture Full information

OLS

Figure 1: Density plots of the three estimators for β

with k explanatory variables which are unobserved at the individual level

yi = α +k∑

j=1

βjxij + εi, (5)

where xij are unobserved 0/1 dummy variables. Assume that we have aggregated infor-

mation summarized in k marginal probabilities Pr[xij = 1] = pij. It is straightforward to

extend the proof above to show that OLS estimator for βj where the xij are replaced by

pij is consistent. If we replace xij by pij, the error term becomes

ηi =k∑

j=1

(xij − pij)βj + εi. (6)

Although the OLS estimator is consistent, it is impossible to estimate the variance of ηi,

because the covariance matrix of xi is unknown. As in practice we often only observe the

marginal probabilities Pr[xij = 1] = pij and not the joint probabilities it is not feasible to

estimate these covariances.

To obtain a more efficient estimator for the β parameters we extend in the next section

the mixture approach to more than one explanatory variable. The proposed approach

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uses the information in the individual responses to estimate the unobserved correlations

between the covariates.

3 Model specification

In this section, we generalize the discussion in the previous section in several ways. First,

we relax the assumption that the model for yi is a linear regression model. Secondly, we

allow for m explanatory variables summarized in the m-dimensional vector Xi. Finally,

we allow for other type of explanatory variables like ordered and unordered categorical

variables and continuous variables. The vector of explanatory variables is written as

Xi = (X(1)i

′, X(2)i

′, X(3)i

′, X(4)i

′)′, where X(1)i contains the binary explanatory variables, X

(2)i

the ordered categorical explanatory variables, X(3)i the unordered explanatory variables

and X(4)i the continuous explanatory variables.

We will use the general model specification

yi = g(Xiβ, εi), (7)

where yi is the observed dependent variable, β is an m-dimensional vector with the param-

eters of interest, εi is a random term, and g is some (non)linear function. The distribution

of εi is known and depends on the unknown parameter vector θ.

This general model can be a linear regression model, but also a limited dependent

variable model or any other nonlinear model. If the Xi variables are observed, parame-

ter estimation is usually standard. In our case, the Xi variables are unobserved at the

individual level but we have sample information at some aggregated level. To estimate

the model parameters β and θ we propose a latent variable model to describe the joint

distribution of the Xi variables. Some of the parameters of this latent variable model are

fixed to match the available sample information at the aggregated level. In the following

subsections we describe the latent variable model for the different types of explanatory

variables.

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3.1 Binary explanatory variables

Assume that X(1)i consists of k binary variables. The joint distribution of X

(1)i is discrete

with 2k mass points which sum up to 1. If we observe these 2k − 1 mass points at

some aggregated level, we can follow the mixture approach of Section 2 to estimate the

β parameters. In practice, however, we typically observe the k marginal probabilities

denoted by P(1)i = (p

(1)i1 , . . . , p

(1)ik )′. Romeo (2005) proposes a method to estimate the joint

discrete distribution from the marginal probabilities. However, he assumes that the joint

distribution is known at an aggregated level. Since we do not have this joint distribution

at an aggregated level, his method is not feasible for our problem at hand.

The k marginal probabilities plus the fact that probabilities sum up to 1 leave us

with 2k − (k + 1) degrees of freedom on the 2k mass points, unless we assume that the

explanatory variables are independent. To facilitate modeling the joint distribution of

X(1)i , we introduce a latent continuous random vector X

(1)∗i = (x

(1)∗i1 , . . . , x

(1)∗ik )′ with

x(1)ij = 1 if x

(1)∗ij > 0

x(1)ij = 0 if x

(1)∗ij ≤ 0

(8)

for i = 1, . . . , N and j = 1, . . . , k, see also Joe (1997) for a similar approach. A convenient

distribution for X(1)∗i is a multivariate normal. The variance of x

(1)∗ij is set equal to 1 for

identification. We impose that the mean of x(1)∗ij equals Φ−1(p

(1)ij ) for j = 1, . . . , k and

i = 1, . . . , N , where Φ denotes the CDF of the standard normal distribution. It holds that

Pr[x(1)ij = 1] = Pr[x

(1)∗ij > 0] = Φ(Φ−1(p

(1)ij )) = p

(1)ij , and hence these restrictions match the

marginal distribution of the X(1)i variables. Hence, we assume that

X(1)∗i ∼ N

(Φ−1(P

(1)i ), Ω1

), (9)

where Ω1 is a k × k positive definite symmetric matrix with ones on the diagonal. This

leaves us with 12k(k − 1) free parameters, the sub-diagonal elements of Ω1. Although

we loose some flexibility by assuming this structure, the correlation parameters do get

an intuitive interpretation as they are related to correlations between the explanatory

variables. The model for X(1)i is in fact a multivariate probit [MVP] model, see Ashford

and Sowden (1970); Amemiya (1974); Chib and Greenberg (1998). In our case however,

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the aggregated data provides restrictions on the intercepts and only the sub-diagonal

elements of Ω1 have to be estimated.

3.2 Ordered categorical explanatory variables

The setup for the binary variables can easily be extended to ordered categorical variables.

If we have one ordered categorical variable with r categories, the X(2)i vector in (7) contains

r−1 0/1 dummies, leaving one category, say the last one, as a reference category. Denote

the r − 1 dummies by x(2)i1 , . . . , x

(2)ir−1. We typically observe the marginal distribution of

the categories at some aggregated level which we denote by the r probabilities P(2)i =

(p(2)i1 , . . . , p

(2)ir )′.

If we only have one ordered categorical explanatory variable in our model, we can use

the simple mixture approach in Section 2 to estimate the effects of the r categories. In

practice, one usually has a combination of several binary and ordered categorical variables

and one has to deal with correlation between these variables. To describe correlations be-

tween several categorical variables, it is convenient to introduce a normal distributed

random variable x(2)∗i to describe the distribution of the categorical variable in the follow-

ing way

x(2)i1 = 1 if x

(2)∗i ≤ qi1 and x

(2)i1 = 0 otherwise

x(2)i2 = 1 if qi1 < x

(2)∗i ≤ qi2 and x

(2)i2 = 0 otherwise

...

x(2)ir−1 = 1 if qir−2 < x

(2)∗i ≤ qir−1 and x

(2)ir−1 = 0 otherwise.

(10)

For identification we impose that the variance of x(2)∗i is 1 such that

x(2)∗i ∼ N (0, 1) . (11)

To match sample probabilities P(2)i , the limit points qi1 . . . qir−1 are set equal to

qij = Φ−1

(j∑

l=1

p(2)il

), i = 1, . . . , N, j = 1, . . . , r − 1. (12)

The proposed model for X(2)i is in fact the ordered probit model of Aitchison and Silvey

(1957).

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The equations (10)–(12) provide the latent variable model for the case of one ordered

categorical explanatory variable. In case one has more categorical variables one can easily

extend the current solution with more latent x(2)∗ij variables and allow them to correlate

using a covariance matrix Ω2 with ones on the diagonal. One can also correlate the

resulting X(2)∗i variables with the latent random variables for the binary variables X

(1)∗i

to describe correlations between binary and ordered categorical explanatory variables.

3.3 Unordered categorical explanatory variables

One may also encounter an explanatory variable which is categorical with, say, r cat-

egories, but that there is no natural ordering. We assume here that an individual can

only belong to one category. If (s)he can belong to several categories one can apply the

approach in Section 3.1. To model the effects of such a variable on yi we include r−1 0/1

dummy variables x(3)i1 , . . . , x

(3)ir−1 in X

(3)i , leaving the rth category as reference. We observe

the marginal probabilities of the r categories at some aggregate level which we denote by

P(3)i = (p

(3)i1 , . . . , p

(3)ir )′.

To deal with the unordered categorical variable we build upon the multinomial probit

[MNP] literature, see, for example, Hausman and Wise (1978) and Keane (1992). We

introduce r − 1 normally distributed variables X(3)∗i = (x

(3)∗i1 , . . . , x

(3)∗ir−1) with

x(3)i1 = 1 if x

(3)∗i1 > max(x

(3)∗i2 , . . . , x

(3)∗ir−1, 0) and x

(3)i1 = 0 otherwise

...

x(3)ir−1 = 1 if x

(3)∗ir−1 > max(x

(3)∗i1 , . . . , x

(3)∗ir−2, 0) and x

(3)ir−1 = 0 otherwise,

(13)

which means that x(3)i1 = . . . = x

(3)ir−1 = 0 if max(x

(3)∗i1 , . . . , x

(3)∗ir−1) ≤ 0. Hence, the vector

X(3)∗i correspond exactly to the utility differences in MNP models. The distribution of

X(3)∗i is given by

x(3)∗i1...

x(3)∗ir−1

∼ N

µ(3)∗i1...

µ(3)∗ir−1

,

1 12

· · · 12

12

. . . . . ....

.... . . . . . 1

212· · · 1

21

, (14)

where µ(3)∗i = (µ

(3)∗i1 , . . . , µ

(3)∗ir−1)

′ represents the mean of X(3)∗i . The covariance matrix

has the same correlation structure as in an MNP model with an identity matrix for the

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individual utilities. The observed probabilities imply r−1 restrictions on the distribution

parameters of X(3)∗i . To match the sample data with the model we have to solve µ

(3)∗i

from

Pr[x(3)∗i1 > x

(3)∗i2 , . . . , x

(3)∗i1 > x

(3)∗ir−1, x

(3)∗i1 > 0] = p

(3)i1

... (15)

Pr[x(3)∗ir−1 > x

(3)∗i1 , . . . , x

(3)∗ir−1 > x

(3)∗ir−2, x

(3)∗ir−1 > 0] = p

(3)ir−1

Pr[x(3)∗i1 ≤ 0, . . . , x

(3)∗ir−1 ≤ 0] = p

(3)ir .

Note that the last restriction is automatically satisfied if the first r − 1 restrictions hold.

Unfortunately, there is no closed form expression for the probabilities from the LHS of

(15) and hence we have to use numerical methods. If r is small, numerical integration

techniques can be used to evaluate the probabilities. For larger values of r the probabilities

can be evaluated using the Stern simulator (Stern, 1992) or the Geweke-Hajivassiliou-

Keane [GHK] simulator (Borsch-Supan and Hajivassiliou, 1993; Keane, 1994). The values

of µ(3)∗i can be found using a numerical solver. Notice that the values of µ

(3)∗i have to be

determined only once before parameter estimation.

The equations (13) and (14) provide the latent variable model in case of one unordered

categorical explanatory variable. In case one has more categorical variables one can easily

extend the current solution in a similar way as discussed before. One can also correlate

the X(3)∗i variables with the X

(1)∗i and X

(2)∗i variables in a straightforward manner.

3.4 Continuous explanatory variables

It may also be the case that the unobserved explanatory variable x(4)i is a continuous vari-

able. Even when the x(4)i variable is continuous, the known distribution of the aggregated

data P(4)i is usually discrete. One may consider this variable as an ordered categorical

variable. However, Breslaw and McIntosh (1998) argue that adding 0/1 dummies to (7)

may not be the best option because it can lead to biased results. Instead, they opt for

imputing the real continuous variable to the response equation (7). Using the approach

in Section 3.2 this implies that we have to add the underlying latent variable for x(4)i , that

is x(4)∗i or a function of x

(4)∗i to X

(4)i in (7) instead of a set of 0/1 dummies. Note that the

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assumption of a normal distribution may be harmful if the real distribution is different.

It is of course also possible to assume another distributions for the latent variable x(4)∗i .

Correlating the X(4)∗i variable with X

(1)∗i , X

(2)∗i and X

(3)∗i can be done in a straightforward

manner.

To summarize this section. The explanatory variables Xi which are missing at the

individual level are described by the latent variable X∗i = (X

(1)∗i

′, X(2)∗i

′, X(3)∗i

′, X(4)∗i

′)′.

This latent variable has a multivariate normal distribution. The mean of this distribution

is determined by the marginal probabilities at the aggregate level. The covariance matrix

of X∗i is denoted by Ω. The free elements in this matrix describe the correlations between

the latent variables X∗i . The models for the X∗

i variables together with (7) provide the

complete model specification.

4 Parameter estimation

To estimate the model parameters of the model proposed in the previous section, we can

choose for maximum likelihood or a Bayesian approach. In this section we discuss both

approaches and their relative merits.

We first derive the likelihood function. Let the density function of the data yi for the

model in (7) conditional on the missing variables Xi be given by

f(yi|Xi; β, θ), (16)

where β and θ denote the model parameters. To derive the unconditional density of yi

we have to sum over all possible values of Xi, which we will denote by the set χ. Hence,

the density of yi given the observed marginal probabilities Pi is given by

f(yi|Pi; β, θ, ρ) =∑Xi∈χ

Pr[Xi|Pi; ρ]f(yi|Xi; β, θ), (17)

where Pr[Xi|Pi; ρ] denotes the probability of observing Xi given the data at the aggregated

level which we denote by Pi = (P(1)′i , P

(2)′i , P

(3)′i , P

(4)′i )′. These probabilities depend on

the unknown parameters ρ which summarizes the free elements of the covariance matrix

of the latent variables Ω, as discussed in the previous section. Hence, the log likelihood

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function is given by

L(y|P ; β, θ, ρ) =N∑

i=1

ln f(yi|Pi; β, θ, ρ), (18)

where y = (y1, . . . , yN)′ and P = (P1, . . . , PN)′. The parameters β, θ and ρ have to be

estimated from the data.

4.1 Maximum likelihood estimation

A maximum likelihood estimator can be obtained by maximizing the log likelihood func-

tion (18) with respect to (β, θ, ρ). To evaluate the log likelihood function we need to

evaluate the probabilities Pr[Xi|Pi; ρ] . Unfortunately, there is no closed form expression

to compute these probabilities. For small dimensions one may use numerical integration

techniques but in general we have to use simulation methods to evaluate the probabilities.

This implies that we end up with a Simulated Maximum Likelihood [SML] estimator, see

Lerman and Manski (1981). The probabilities Pr[Xi|Pi; ρ] can be evaluated using the

Stern simulator (Stern, 1992) or the GHK simulator (Borsch-Supan and Hajivassiliou,

1993; Keane, 1994).

The SML estimator is consistent if the number of observations and the number of

simulations goes to infinity. Given the literature on SML in MNP models (see for example,

Geweke et al., 1994), one may expect that obtaining an accurate values of the probabilities

Pr[Xi|Pi; ρ] is computationally intensive, especially when the dimension of the latent X∗i is

large and/or the number of observations N is large. Note that the number of probabilities

Pr[Xi|Pi; ρ] which need to be evaluated grows exponentially with the number of variables

in Xi.

4.2 Bayesian analysis

The model can also be analyzed in a Bayesian framework. To obtain posterior results

for the model parameters, we propose a Gibbs sampler (Geman and Geman, 1984) with

data augmentation, see Tanner and Wong (1987). The latent X∗i variables are simulated

along side the model parameters (β, θ, ρ). The main advantage of this Bayesian approach

is that it does not require the evaluation of the complete likelihood function. If suffices

to evaluate the likelihood function conditional on the latent X∗i which determine Xi.

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We focus in this section on the sampling of the latent variable X∗i . We assume that if

we know the X∗i and hence the Xi variables, a MCMC sampling scheme to simulate from

the posterior distribution of the model parameters β and θ is available. Hence, we do

not discuss simulating from the full conditional distribution of β and θ as this is model

specific. We do however discuss simulating from the full conditional distribution of the ρ

parameters as this is part of the model for the latent variable X∗i .

4.2.1 Sampling of XXX∗∗∗iii

To sample X∗i we have to derive its full conditional density which is given by

f(X∗i |yi, Pi; β, θ, ρ) ∝ f(yi|Xi; β, θ)f(X∗

i |Pi; ρ)f(ρ), (19)

where f(yi|Xi; β, θ) is given in (16), where f(X∗i |Pi; ρ) denotes the density of X∗

i implied

by the latent variable model for X∗i , and where f(ρ) denotes the prior distribution for ρ.

Given the structure of the latent variable model, X∗i is a multivariate normal distribution

with a mean µi which depends on Pi and a covariance matrix Ω where the free elements

are denoted by ρ, that is,

f(X∗i |Pi; ρ) = φ(X∗

i ; µi(Pi), Ω(ρ)), (20)

where φ denotes the multivariate normal density function. Sampling the complete X∗i

vector at once is very difficult. Therefore, we sample the individual elements of X∗i

separately from their full conditional distribution. Let us consider the jth element of X∗i

denoted by x∗ij. The full conditional density of x∗ij is given by

f(x∗ij|yi, X∗i,−j, Pi; β, θ, ρ) ∝ f(yi|xij, Xi,−j; β, θ)f(x∗ij|X∗

i,−j, Pi; ρ), (21)

where X∗i,−j and Xi,−j denote the vector X∗

i and Xi without x∗ij and xi,−j, respectively.

The full conditional density of x∗ij consists of two parts. The second part f(x∗ij|X∗i,−j, Pi; ρ)

is the conditional density of one of the elements of X∗i which is of course a normal density

with known mean, say, µij, and variance, say, s2j , which are functions of µi(Pi) and Ω(ρ).

The first part f(yi|xij, Xi,−j; β, θ) is a function of Xi and can take a discrete number of

values depending on the value of x∗ij. If we for simplicity assume that x∗ij corresponds to a

15

binary explanatory variable, xij can take two values depending one whether x∗ij is larger

or smaller than 0.

To sample from the full conditional posterior distribution of x∗ij we use the inverse

CDF technique. In Appendix A we derive the inverse CDF of x∗ij. The simulation scheme

for x(1)∗ij corresponding to the binary variable x

(1)ij can be summarized as follows

1. Draw u ∼ U(0, 1)

2. Set

x(1)∗ij =

Φ−1(

uciki0

)sj + µij if u ≤ ciki0Φ

(−µij

sj

)

Φ−1(

uciki1

+ ki1−ki0

ki1Φ

(−µij

sj

))sj + µij if u > ciki0Φ

(−µij

sj

) ,

where ki0 = f(yi|x(1)ij = 0, Xi,−j; β, θ), ki1 = f(yi|x(1)

ij = 1, Xi,−j; β, θ), and ci is the

integrating constant of the full conditional distribution of x(1)∗ij given by

ci =

((ki0 − ki1)Φ

(−µij

sj

)+ ki1

)−1

. (22)

Appendix A also provides the derivations of the inverse CDF in case x∗ij is associated

with an ordered or an unordered categorical variable.

4.2.2 Sampling of ρρρ

To complete the Gibbs sampler, we need to sample the parameters in ρ from their full con-

ditional posterior distribution. The vector ρ contains the free elements of the covariance

matrix of X∗i which is denoted by Ω. As discussed in Section 3, identification requires

several restrictions on the covariance matrix Ω. For example, all diagonal elements of Ω

are equal to 1 and hence Ω is a correlation matrix. Furthermore, the correlations between

elements of the same unordered categorical variable are set equal to 1/2. Hence, the full

conditional distribution of Ω is not an inverted Wishart distribution.

There are some algorithms to draw a correlation matrix, see, for example, Chib and

Greenberg (1998), Manchanda et al. (1999), and Liechty et al. (2004). In this paper we

follow Barnard et al. (2000). They suggest sampling one correlation at a time from their

full conditional posterior distribution using a griddy-Gibbs sampler, see Ritter and Tanner

(1992).

16

Suppose we want to draw the jth correlation in ρ denoted by ρj. Denote the vector

ρ without ρj as ρ−j. Furthermore, let X∗ = (X∗1 , . . . , X

∗N)′ and µ = (µ1, . . . , µN)′, where

µi denotes the mean of X∗i and is a function of Pi, for i = 1, . . . , N . The full conditional

posterior density of ρj is given by

f(ρj|ρ−j, X∗) ∝ f(X∗|ρj, ρ−j)f(ρj|ρ−j)

∝N∏

i=1

φ(X∗i ; µi, Ω(ρj, ρ−j))f(ρj|ρ−j)

∝ |Ω(ρj, ρ−j)|−N2 exp

[−1

2tr((X∗ − µ)′(X∗ − µ)Ω(ρj, ρ−j)

−1)

]f(ρj|ρ−j).

(23)

Barnard et al. (2000) show how to determine the range of values for which ρj leads to a

positive definite matrix. Within this range we can define a set of grid points to evaluate

the kernel (23) for the griddy-Gibbs sampler.

As correlations in Ω which are related to the jth explanatory variable are not identified

if βj = 0, we have to impose an informative prior for the parameters in ρ. We use a

(truncated) standard normal prior for the parameters in ρ, that is,

f(ρj) ∝ exp(ρ2j). (24)

Hence, we concentrate the probability mass around zero.

5 Application

To illustrate our approach, we consider in this section an application where we analyze

the characteristics of households who donate to a large Dutch charity in the health sector.

Households receive a direct mailing form the charity with a request to donate money. The

household may not respond and donate nothing or respond and donate a positive amount.

We have no information about the characteristics of the households apart from their zip

code. At the zip-code level we know aggregated household characteristics.

Our sample contains 10,000 households which are randomly selected. The mailing took

place in February 1997. The response rate is 39.4%. The average donation is 3.21 euros

and the average donation conditional on response is 8.15 euros. We match these data

with aggregated data at the zip-code level (4 digits) from Statistics Netherlands (CBS).

17

Table 2: Available explanatory variables at the zip-code levelVariable Type DescriptionChurch Binary Goes to church every week

Not-active Binary Not active in labor force

Reference: Family with kidsSingle Unordered (3 cat.) Lives aloneFamily no kids Unordered (3 cat.) Family without kids

Reference: Average incomeIncome low Ordered (3 cat.) Income in lowest 40% nationallyIncome high Ordered (3 cat.) Income in highest 20% nationally

Reference: Age between 25 and 44Age 0-24 Ordered (4 cat.) Age between 0 and 24Age 45-64 Ordered (4 cat.) Age between 45 and 64Age 65+ Ordered (4 cat.) Age over 65

Urbanization Observed Measure for the degree of urbanization

Table 2 shows the relevant aggregated data at the zip-code level. As can be observed

from the table we have aggregated data for different types of explanatory variables, that

is, for binary, unordered, and ordered categorical variables. Note that we only know

urbanization level at the zip-code level. As it is the same for each individual in the

zip-code region, this variable is treated as an observed variable.

To describe donating behavior we consider a censored regression (Tobin, 1958) because

the donated amount is censored at 0. We use the log of (1 + amount) as dependent variable

which leads to the following model specification

ln(1 + yi) = x′iβ + εi if x′iβ + εi > 0ln(1 + yi) = 0 if x′iβ + εi ≤ 0

(25)

with εi ∼ N(0, σ2). As explanatory variables we take the variables displayed in Table 2.

To estimate the effects of the covariates on response, we use two approaches . First,

we follow the simple regression approach of Section 2, which means that we replace the

unknown household characteristics by their sample averages at the zip-code level. The

parameters of (25) are estimated using ML. Although we have only shown in Section 2 that

OLS in a linear regression model provides consistent estimates, simulations suggest that

this results carries over to the ML estimator in a censored regression model. Secondly, we

use the mixture approach to estimate the censored regression parameters, where we opt for

18

Table 3: Posterior means, posterior standard deviations, and HPDregions of the model parameters for the mixture approach togetherwith ML results for the simple approach

mixture approach simple approachmean st.dev. 95% HPD ML s.e.a

Intercept -0.36 0.008 (-0.37,-0.34) -2.19 0.82Urbanization -0.00 0.008 (-0.02, 0.01) 0.38 0.34Church 0.33 0.003 ( 0.32, 0.34) 0.15 0.24Not-active 0.70 0.005 ( 0.69, 0.71) 0.24 0.70Single -1.14 0.010 (-1.16,-1.12) 0.57 0.56Family no kids 2.07 0.008 ( 2.06, 2.09) 2.82 1.17Income low -1.83 0.005 (-1.84,-1.82) -1.30 1.11Income high 0.59 0.003 ( 0.58, 0.59) 1.18 0.83Age 0-24 0.73 0.006 ( 0.72, 0.74) 2.62 1.35Age 45-65 -0.57 0.003 (-0.58,-0.57) -0.35 1.24Age 65+ 0.22 0.004 ( 0.21, 0.23) 1.60 1.00σ 0.06 0.001 ( 0.06, 0.06) 2.26 0.02a Heteroskedasticity consistent standard errors, see White (1982).

a Bayesian approach. We use an uninformative prior for β and σ2, that is, p(β, σ2) ∝ σ−2

and the informative prior (24) for the ρ parameters. We use a total of 110,000 draws, of

which the first 50,000 were used as burn-in period. Furthermore, we only used every 12th

draw to obtain an approximately random sample of 5,000 draws.

Table 3 displays the estimation results for both approaches. It is clear from the table

that the posterior standard deviations of the mixture approach are much smaller than

the standard errors of the ML estimator, where the unknown household characteristics

are replaced by their sample averages at the zip-code level. Although the number of ob-

servations is very high, the estimated standard errors are very large. This illustrates the

huge efficiency gain of using our method. This efficiency gain enables us to identify more

significant influences from household characteristics. When using the simple approach,

only Family no kids is identified as having a significant impact on the donating behavior.

But, using our mixture approach it becomes clear that many other household character-

istics also influence this decision. Being Religious and not being active in the labor force

has a positive effect. Note that latter group also contain retired people. Being single has

a negative effect, while families without children tend to donate more. Household with

19

Table 4: Posterior means of the correlations between unobserved variableswith posterior standard deviations in parenthesis

Church Not-active

Single Familyno kids

Income Age

Church 1(-)

Not-active -0.19 1(0.05) (-)

Single -0.24 0.68 1(0.05) (0.02) (-)

Family -0.23 0.15 0.50 1no kids (0.03) (0.03) (-) (-)Income -0.08 -0.25 0.43 0.75 1

(0.04) (0.03) (0.02) (0.02) (-)Age -0.12 -0.82 -0.24 0.23 0.67 1

(0.06) (0.02) (0.03) (0.03) (0.02) (-)

higher income tend to donate more, while the effect of age is nonlinear. The highest pos-

terior density [HPD] interval shows that urbanization grade has no influence on donating

behavior.

Table 4 displays the estimated correlation matrix of X∗i . The diagonal elements are

put equal to 1 for identification. The correlation between the variables Single and Family

no kids is fixed at 1/2, because they belong to the same unordered categorical variable.

Most of the posterior means of the correlations are more than two times larger than

their posterior standard deviation, illustrating the importance of our approach. The

correlations usually have the expected sign. For example, there is a negative correlation

between being not active and income, and a negative correlation between being single and

age.

6 Conclusions

In this paper we have developed a new approach to estimate the effects of explanatory

variables on individual response where the response variable is observed at the individual

level but the explanatory variables are only observed at some aggregated level. This

approach can, for example, be used if information about individual characteristics is

only available at the zip-code level. To solve the limited data availability, we extend

20

the model describing individual responses with a latent variable model to describe the

missing individual explanatory variables. The latent variable model is of the probit type

and matches the sample information of the explanatory variables at the aggregated level.

Parameter estimates for the effects of the explanatory variables in the individual response

model can be obtained using maximum likelihood or a Bayesian approach.

A simulation study shows that our new approach clearly outperforms a standard ap-

proach in efficiency. The efficiency loss which is due to aggregation is about 50% smaller

than for the standard method. We illustrated the merits of our approach by estimat-

ing the effects of the household characteristics on donating behavior to a Dutch charity.

For this application we used data of donating behavior at the household level, while the

covariates were only observed at the zip-code level.

There are several ways for future research. It may be interesting to investigate whether

the proposed method can be used to deal with nonresponse in survey data. Another topic

for future research is to consider the complement case where explanatory variables are

observed at the individual level but that the response variable is only observed at some

aggregated level.

21

A Derivation of full conditional distributions

In this appendix we give a short derivation for the full conditional posterior distributions in

case the missing explanatory variable is a binary variable, an ordered categorical variable

and an unordered categorical variable. As starting point we take the general form of the

full conditional density of x∗ij given in (21).

A.1 Binary variable

In case xij is a binary variable, the posterior distribution of x(1)∗ij can be summarized by

f(x(1)∗ij |yi, X

∗i,−j, Pi; β, θ, ρ) =

ci(ki0φ(x(1)∗ij ; µij, s

2j)I[x

(1)∗ij ≤ 0] + ki1φ(x

(1)∗ij ; µij, s

2j)I[x

(1)∗ij > 0]), (26)

where X∗i−j denotes X∗

i without x(1)∗ij , ki0 = f(yi|x(1)

ij = 0, Xi,−j; β, θ), ki1 = f(yi|x(1)ij =

1, Xi−j; β, θ), φ(x; m, s) denotes the normal density function with mean m and variance

s evaluated in x, µij denotes the conditional mean of x(1)∗ij |X∗

i,−j, ρ, and where s2j denotes

the conditional variance of x(1)∗ij |X∗

i,−j, ρ. The integrating constant ci follows from

c−1i =

∫ 0

−∞ki0φ(x

(1)∗ij , µij, s

2j)dx

(1)∗ij +

∫ ∞

0

ki1φ(x(1)∗ij , µij, s

2j)dx

(1)∗ij

c−1i = ki0Φ

(−µij

sj

)+ ki1

(1− Φ

(−µij

sj

))

ci =

((ki0 − ki1)Φ

(−µij

sj

)+ ki1

)−1

, (27)

where Φ denotes the standard normal distribution function. Hence, the full conditional

density of x(1)∗ij is given by

f(x(1)∗ij |yi, X

∗i,−j, Pi; β, θ, ρ) =

ciki0φ(x

(1)∗ij , µij, sj) if x

(1)∗ij ≤ 0

ciki1φ(x(1)∗ij , µij, sj) if x

(1)∗ij > 0

(28)

and the distribution function reads

F (x(1)∗ij |yi, X

∗i,−j, Pi; β, θ, ρ) =

ciki0Φ

(x(1)∗ij −µij

sj

)if x

(1)∗ij ≤ 0

ci

[(ki0 − ki1)Φ

(−µij

sj

)+ ki1Φ

(x(1)∗ij −µij

sj

)]if x

(1)∗ij > 0.

(29)

22

To draw from this distribution we use the inverse CDF technique, which means that

we draw u ∼ U(0, 1) and solve x(1)∗ij from F (x

(1)∗ij |yi, X

∗i,−j, Pi; β, θ, ρ) = u. The inverse

function of (29) is given by

x(1)∗ij (u) =

Φ−1(

uciki0

)sj + µij if u ≤ ciki0Φ

(−µij

sj

)

Φ−1(

uciki1

+ ki1−ki0

ki1Φ

(−µij

sj

))sj + µij if u > ciki0Φ

(−µij

sj

).

(30)

A.2 Ordered categorical variable

For an ordered categorical variable with r categories, we have to sample the variable x(2)∗ij .

If we assume that r is the reference category, the x(2)∗ij variable determines the r − 1 0/1

dummy variables x(2)i1 , . . . , x

(2)ir−1. Let P

(2)i = (p

(2)i1 , . . . , p

(2)ir )′ denote the observed marginal

probabilities that the individual belongs to the r categories. The threshold levels qit

are equal to Φ−1(∑t

l=1 p(2)il ) for i = 1, . . . , N and t = 1, . . . , r − 1. Let µij denote the

conditional mean of x(2)∗ij |X∗

i,−j, ρ in the latent model and let s2j denote the conditional

variance of x(2)∗ij |X∗

i,−j, ρ, where X∗i,−j denotes X∗

i without x(2)∗ij .

Sampling of x(2)∗ij proceeds in the same way as for the binary variables except for the

fact that there are now r possible values for f(yi|Xi; β, θ) instead of only 2. The r values

for f(yi|Xi; β, θ) are given by

ki1 = f(yi|Xi,−j, x(2)i1 = 1, x

(2)i2 = 0, . . . , x

(2)ir−1 = 0; β, θ)

...

kir−1 = f(yi|Xi,−j, x(2)i1 = 0, . . . , x

(2)ir−2 = 0, x

(2)ir−1 = 1; β, θ)

kir = f(yi|Xi,−j, x(2)i1 = 0, . . . , x

(2)ir−1 = 0; β, θ),

where Xi,−j denotes Xi without x(2)i1 , . . . , x

(2)ir−1. The integrating constant now equals

ci =

(r−1∑t=1

(kit − kit+1)Φ

(qit − µij

sj

)+ kir

)−1

. (31)

Straightforward derivation leads to following inverse CDF

x(2)∗ij (u) =

Φ−1(

uciki1

)sj + µij if u ≤ u1

Φ−1[

uciki2

− ki1−ki2

ki2Φ

(qi1−µij

sj

)]sj + µij if u1 < u ≤ u2

......

Φ−1[

ucikir

−∑r−1t=1

kit−kit+1

kirΦ

(qit−µij

sj

)]sj + µij if ur−1 < u,

(32)

23

where

u1 = ciki1Φ

(qi1 − µij

sj

)

u2 = ciki2Φ

(qi2 − µij

sj

)+ ci(ki1 − ki2)Φ

(qi1 − µij

sj

)

...

ur−1 = cikir−1Φ

(qir−1 − µij

sj

)+ ci

r−2∑t=1

(kit − kit+1)Φ

(qit − µij

sj

).

A.3 Unordered categorical variable

For an unordered categorical variable with r categories, we add r − 1 0/1 dummies in

X(3)i , say, x

(3)i1 , . . . , x

(3)ir−1. The r− 1 normal distributed random variables which belong to

this unordered categorical variable are denoted by (x(3)∗i1 , . . . , x

(3)∗ir−1)

′.

Suppose that we want to sample x(3)∗ij from its full conditional posterior distribution.

The full conditional posterior density is given by (21). The second part f(x(3)∗ij |X∗

i,−j, Pi; ρ),

where X∗i−j denotes X∗

i without x(3)∗ij , is a normal density with known mean, say µij, and

standard deviation, say sj. Conditional on X∗i,−j the first part f(yi|xij, Xi,−j; β, θ) can

take two values. Define x(3)∗il = max(x

(3)∗i1 , . . . , x

(3)∗ij−1, x

(3)∗ij+1, . . . , x

(3)∗ir−1). The two possible

values for f(yi|xij, Xi,−j; β, θ) are given by

ki0 = f(yi|x(3)i1 = 0, . . . , x

(3)il−1 = 0, x

(3)il = 1, x

(3)il+1 = 0, . . . x

(3)ir−1 = 0, Xi,−j; β, θ)I[x

(3)∗il > 0] +

f(yi|xi1 = 0, . . . , xir−1 = 0, Xi,−j; β, θ)I[x(3)∗il ≤ 0]

ki1 = f(yi|x(3)i1 = 0, . . . , x

(3)ij−1 = 0, x

(3)ij = 1, x

(3)ij+1 = 0, . . . x

(3)ir−1 = 0, Xi,−j; β, θ),

where Xi,−j denotes Xi without x(3)i1 , . . . , x

(3)ir−1. The full conditional posterior density of

x(3)∗ij is given by

ci(ki0φ(x(3)∗ij ; µij, sj)I[x

(3)∗ij ≤ max(x

(3)∗il , 0)] + ki1φ(x

(3)∗ij ; µij, sj)I[x

(3)∗ij > max(x

(3)∗il , 0)]).

(33)

The integrating constant ci is given by

ci =

((ki0 − ki1)Φ

(max(x

(3)∗il , 0)− µij

sj

)+ ki1

)−1

. (34)

24

Straightforward derivation leads to the following inverse CDF

x(3)∗ij (u) =

Φ−1(

uciki0

)sj + µij if u ≤ ciki0Φ

(max(x

(3)∗il ,0)−µij

sj

)

Φ−1

(u

ciki1+ ki1−ki0

ki1Φ

(max(x

(3)∗il ,0)−µij

sj

))sj + µij if u > ciki0Φ

(max(x

(3)∗il ,0)−µij

sj

).

(35)

25

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